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Prices in Mixed Cost Allocation Problems

E. Calvo, J.C. Santos
2001 Games and Economic Behavior  
., joint cost problems in which there are two types of heterogeneous outputs : Divisible and indivisible ones. The Aumann-Shapley price mechanism is extended to this setting.  ...  Owen (1972) de nes the concept of multilinear extension of a TU-game: this extension can be seen as a continuum game, thereby the Shapley value turns out to be the Aumann-Shapley value of its multilinear  ...  In Calvo and Santos (1997) an asymptotic relation was shown between the multichoice values of a sequence of multichoice games which converge to the continuum game and the value of this continuum game  ... 
doi:10.1006/game.2000.0840 fatcat:xzl5wm3kyvemtjaifjkuvknboi

On the serial cost sharing rule

M. J. Albizuri, J. C. Santos, J. M. Zarzuelo
2003 International Journal of Game Theory  
We show that this solution can be computed by applying the Shapley value to an appropriate TU game and we present a probabilistic formula.  ...  We also define for cost allocation problems a multilinear function in order to obtain the serial cost sharing method as Owen (1972) did for the Shapley value in the cooperative TU context.  ...  These games were introduced by Hsiao and Raghavan (1992) . For these games, di¤erent extensions of the Shapley value have been studied (this literature is surveyed in Calvo and Santos, 2000) .  ... 
doi:10.1007/s001820300129 fatcat:rigkoufv3rfyrdrt5bezmyrxnu

A value for multichoice games

Emilio Calvo, Juan Carlos Santos
2000 Mathematical Social Sciences  
Also, we show that the Aumann-Shapley value for continuum games can be obtained as the limit of multichoice values for admissible convergence sequences of multichoice games.  ...  Cores and related solution concepts for multi-choice games. ZOR-Mathematical Methods of Operations Research 41, 289-311].  ...  In Van Den Nouweland (1993) an example is given of a multichoice game for which this value is not equal to any of the values of Hsiao and Raghavan; and other alternative proposals for the multichoice  ... 
doi:10.1016/s0165-4896(99)00054-2 fatcat:xuitth42ufchfjwpg4trascl6a

Bisemivalues for bicooperative games

Margarita Domènech, José Miguel Giménez, María Albina Puente
2018 Optimization  
For the particular case of (p, q)-bisemivalues, a computational procedure in terms of the multilinear extension of the game is given.  ...  We introduce bisemivalues for bicooperative games and we also provide an interesting characterization of this kind of values by means of weighting coefficients in a similar way as it was given for semivalues  ...  As is well known, both the Shapley and Banzhaf values of any cooperative game v can be obtained from its multilinear extension.  ... 
doi:10.1080/02331934.2018.1432611 fatcat:cm5ryakttvdlpa4jtnxgtvdsp4

Page 4254 of Mathematical Reviews Vol. , Issue 2004e [page]

2004 Mathematical Reviews  
For these games different extensions of the Shapley value have been studied.  ...  to efficient, linear, and symmetric values for TU-games.  ... 

A Multilinear Sampling Algorithm to Estimate Shapley Values [article]

Ramin Okhrati, Aldo Lipani
2020 arXiv   pre-print
Castro et al. [1] developed a sampling algorithm, to estimate Shapley values. In this work, we propose a new sampling method based on a multilinear extension technique as applied in game theory.  ...  Shapley values are great analytical tools in game theory to measure the importance of a player in a game.  ...  Jones and Wilson [12] present a multilinear extension technique, which is then applied to multichoice games in the context of Shapley values using the central limit theorem. Michalak et al.  ... 
arXiv:2010.12082v1 fatcat:6ujpphi5mnhprnkctkcyhcbkem

On importance indices in multicriteria decision making

Michel GRABISCH, Christophe LABREUCHE, Mustapha RIDAOUI
2019 European Journal of Operational Research  
Our analysis first considers discrete models, which are seen to be equivalent to multichoice games.  ...  We provide several axiomatizations for these importance indices, extend them to the continuous case, and finally illustrate them with examples (classical simple models and a example of discomfort evaluation  ...  We note that a corresponding result holds for the MLE (multilinear extension of Owen (1972) ) model and the Banzhaf value (Banzhaf, 1965 ) (see, again, Grabisch et al. (2000) ).  ... 
doi:10.1016/j.ejor.2019.02.035 fatcat:zxu42vzsdndavgathgj36455pi

A Multilinear Sampling Algorithm to Estimate Shapley Values

Ramin Okhrati, Aldo Lipani
2021 2020 25th International Conference on Pattern Recognition (ICPR)  
Castro et al. [1] developed a sampling algorithm, to estimate Shapley values. In this work, we propose a new sampling method based on a multilinear extension technique as applied in game theory.  ...  Shapley values are great analytical tools in game theory to measure the importance of a player in a game.  ...  Jones and Wilson [12] present a multilinear extension technique, which is then applied to multichoice games in the context of Shapley values using the central limit theorem. Michalak et al.  ... 
doi:10.1109/icpr48806.2021.9412511 fatcat:dwvzbkcaznhgdcklb6ckamwgmq

The interaction transform for functions on lattices

Fabien Lange, Michel Grabisch
2009 Discrete Mathematics  
It generalizes the notion of interaction defined for players modeled by games, by considering functions defined on distributive lattices.  ...  A general definition of the interaction transform is provided, as well as the construction of operators establishing transforms between games, their Möbius transforms and their interaction indices.  ...  In [14] , we proposed an axiomatization of the Shapley value for multichoice games, where the obtained formula is also the one given above (all the L i 's are completely ordered).  ... 
doi:10.1016/j.disc.2008.12.007 fatcat:b7qbm6c7r5d3jpmdyfxz7ysxj4

An axiomatic characterization of the potential decisiveness index

Josep Freixas, Montserrat Pons
2015 Journal of the Operational Research Society  
The potential decisiveness index for simple games, which measures the effect that ensuring one positive vote produces in the probability to pass the issue at hand, is a good tool to get the answer.  ...  Let us consider that somebody is extremely interested in increasing the probability of a proposal to be approved by a certain committee and let us assume that for achieving this goal he/she is prepared  ...  [3] , [16] and [21] for successive wider results on simple games. It also deserves interest the extension of these measures for multichoice games or for games with a coalition structure, see e.g  ... 
doi:10.1057/jors.2014.5 fatcat:7tio6lnlp5aepje5rmsgygtuc4

Players indifferent to cooperate and characterizations of the Shapley value

C. Manuel, E. González-Arangüena, R. van den Brink
2012 Mathematical Methods of Operations Research  
In this spirit, the multilinear extension of Owen (1972) also admits and additive decomposition.  ...  , Grabish and Lange (2007) on multichoice games and Khmelnitskaya and Yanovskaya (2007) on games with coalition struture.  ... 
doi:10.1007/s00186-012-0412-7 fatcat:uwpywljpabgtfjiu54jovhhjzy

Players Indifferent to Cooperate and Characterizations of the Shapley Value

Conrado Manuel, Enrique Gonzalez-Aranguena, René van den Brink
2012 Social Science Research Network  
In this spirit, the multilinear extension of Owen (1972) also admits and additive decomposition.  ...  , Grabish and Lange (2007) on multichoice games and Khmelnitskaya and Yanovskaya (2007) on games with coalition struture.  ... 
doi:10.2139/ssrn.2039234 fatcat:vplhlez5pjf5lpz3brfgkn52cq

Evolutionary consequences of behavioral diversity [article]

Alexander J. Stewart, Todd L. Parsons, Joshua B. Plotkin
2016 arXiv   pre-print
Finally, we analyze iterated rock-paper-scissors games, whose non-transitive payoff structure means unilateral control is difficult and zero-determinant strategies do not exist in general.  ...  Iterated games provide a framework to describe social interactions among groups of individuals.  ...  Our analysis has relied on the theory of iterated games and, in particular, on a coordinate system we developed to describe strategies for multichoice games and their effects on long-term payoffs.  ... 
arXiv:1606.01401v1 fatcat:d4qfrhw73ndtjdwh64ztzn2ytm

Using multiple reference levels in Multi-Criteria Decision aid: The Generalized-Additive Independence model and the Choquet integral approaches

Christophe Labreuche, Michel Grabisch
2018 European Journal of Operational Research  
.), but differ on the interpolation means (Lovász extension for the Choquet integral, and multi-linear extension for the GAI model).  ...  In the context of cooperative game theory, this concept is similar to multichoice games [27] . Let us define Q p (N ) = {q ∈ Q p (N ) : q i < p i ∀i ∈ N }.  ...  The only difference is a matter of extension -namely Lovász extension for U ∧ and multi-linear for U Π (see Section 2.2).  ... 
doi:10.1016/j.ejor.2017.11.052 fatcat:rdzfaqqp2zgirpsr2lr3ahb7qq

Page 1365 of Mathematical Reviews Vol. 31, Issue Index [page]

Mathematical Reviews  
.; Tanaka, Tamaki) Construction of a time consistent core in multichoice multistage games. (English summary) Decision theory and related fields (Japanese) (Kyoto, 1997).  ...  (with Samoilenko, Anatoli Mikhailovich) Boundary value problems with parameters for multifrequency oscillatory systems. (Ukrainian. English and Ukrainian summaries) Ukrain. Mat.  ... 
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